Talk:Superparticular ratio

Generalizing the notion of splitting a ratio into superparticular ratios

"Every superparticular ratio can be split into the product of two superparticular ratios." Take 5/4, double everything to get 10/8, then convert that to 10/9 x 9/8. You can likewise triple 5/4 to get 15/12 which is 15/14 x 14/13 x 13/12. And so forth. So in fact every superparticular ratio can be split into the product of N superparticular ratios for any natural number N.

Every delta-2 ratio can be split into the product of two superparticular ratios, because (N+2)/N = (N+2)/(N+1) x (N+1)/N. For example, 5/3 = 5/4 x 4/3. (So in my earlier example where 5/4 becomes 10/8, I simply converted a reduced delta-1 ratio into an unreduced delta-2 ratio.) You can likewise double everything in a delta-2 ratio, converting it into a delta-4 ratio, e.g. 5/3 becomes 10/6. Obviously every delta-4 ratio can be split into 4 delta-1 ratios. Likewise a delta-2 ratio can be tripled to a delta-6 ratio, and split into 6 parts. So in fact every delta-2 ratio can be split into the product of N superparticular ratios, where N is even.

Likewise every delta-3 ratio can be split into the product of N superparticular ratios, where N is threeven. And so forth. -TallKite (talk) 07:21, 27 February 2023 (UTC)

Redundant statement

In the Properties section it says "The logarithmic sum (i.e. product) of two successive superparticular ratios is either a superparticular ratio or a superpartient ratio." And yet every ratio is either a superparticular ratio or a superpartient ratio. -TallKite (talk) 07:53, 1 March 2023 (UTC)